Madness

[Oldschoolmtgnoob]

Does the madness ability let you cast at instant speed if it was discarded as an additional cost of an instant?

[Kaylesh]

Note that madness has two abilities. A replacement ability that lets you exile the card if you discard it, and a triggered ability that lets you cast it from exile, at that time.

An opponent can use an Eldrazi processor such as  {Cryptic Cruiser} to put the card into your graveyard before the triggered ability resolves. In that case, you won't be able to cast it.
(Relevant in upcoming standard)

[General Kiwi]

Note that madness has two abilities. A replacement ability that lets you exile the card if you discard it, and a triggered ability that lets you cast it from exile, at that time.

An opponent can use an Eldrazi processor such as  {Cryptic Cruiser} to put the card into your graveyard before the triggered ability resolves. In that case, you won't be able to cast it.
(Relevant in upcoming standard)

Oh that's huge! Thank you for that

[Remillo]

Also note that discarding to exile is now mandatory with the new rules update.

Note that the mandatory discard into Exile is a small change from previous rules. Before, you could discard a card with madness into your graveyard and skip the whole madness thing. This may be relevant with cards like Jace, Vryn's Prodigy. That dude shows up a lot in this article for a card not in this set.

from Matt Tabak's rules update article

You discard it to exile, then when the trigger resolves you choose whether to cast it or put it into your graveyard.

[Oldschoolmtgnoob]

So let me order-of-operations this bad boy just to make sure I understand.

Cast a spell with an additional/alternative cost of discarding a card.
Choose a card with madness to discard; replacement effect kicks in and it goes to exile.
First spell stacks. Madness ability stacks.

Priority passes (opportunity for processing?)

Choose whether to cast madness card. If chosen to cast, spell stacks.

Priority passes

Madness card resolves

Pass

Original spell resolves.

Getting?

[Remillo]

You got it, man!

And if you discard it as part of the RESOLUTION of a spell or ability instead of part of the cost, the Madness trigger won't resolve until after the effect that caused you to discard it finishes resolving.  This is why if you discard a card to {Jace, Vryn's Prodigy} that /would/ be the fifth card, he won't flip.  The card with madness is still floating in exile when his ability checks the number of cards in your graveyard.